We calculate the 5PN Hamiltonian ab initio starting with the path integral
for classical gravity, for which a post–Newtonian expansion is performed by applying an EFT description
[1]}, [2]}. Methods originally developed for Quantum Field Theories are applied.
About 190.000 Feynman diagrams contribute. They are generated by QGRAF [3]}. Main
parts of the calculation are performed using FORM [4]}. The reduction to a very small set of
master integrals is performed using Crusher [5]} using the integration-by-parts relations
[6]}.
The 5PN potential terms have been calculated by us in Ref. [7]}.We compared
also to the factorizing contributions to the potential terms, which have been obtained in [8]},
version 2, very recently, to which the corresponding subset of our results is agreeing. There we also
described how the singularities in the potential and tail terms
[9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}
are canceling, together with an additional canonical transformation. The whole 5PN calculation is performed
starting in the Lagrange formalism and finally deriving the Hamiltonian.As outlined in detail
in Refs. [17]}, [7]}, the treatment of higher time derivatives has to be
performed without applying the equation of motion. Note, that there are different approaches in the
literature. Also the non–local 5PN contributions were
presented, see also [10]}.
What remained to be calculated beyond the contributions given in [7]} is a series of local tail
contributions. These terms contribute at \(O(\nu )\)  and \(O(\nu ^2)\) . They are due to the 1PN correction
to the electric quadrupole moment, written symbolically, \(E Q_{ij} Q_{ji}\) , with \(E\)  the energy,
[15]}, the octupole moment,
\( E O_{ijk} O_{ijk}\) , the magnetic quadrupole moment, \(E J_{ij} J_{ji}\) , the angular momentum failed tail, \(L_k\varepsilon _{ijk}Q_{il} Q_{jl}\) , with \(\vec{L}\)  the angular momentum, and the memory terms, \(Q_{ij} Q_{jk} Q_{ki}\) , which
have also
been considered in Ref. [16]}.For a definition of the multipole moments see
e.g. [23]}. These multipole contributions are calculated in \(D\)  dimensions starting
with
harmonic coordinates. As has been outlined in [7]} only the electric quadrupole, octupole, and
magnetic quadrupole moments have poles of \(O(1/\varepsilon )\)  and receive logarithmic contributions. All other
multipole
moments can be calculated in \(D=4\)  dimensions and form rational contributions to the 5PN Hamiltonian. We have
performed an independent calculation of these and related contributions and performed a detailed comparison
with [16]}.The result in [16]} for the angular momentum failed tail still needs a sign change, as
communicated to us by the authors.
